Fractal convolution: A new operation between functions

  title={Fractal convolution: A new operation between functions},
  author={Mar{\'i}a A. Navascu{\'e}s and Peter R. Massopust},
  journal={Fractional Calculus and Applied Analysis},
  pages={619 - 643}
Abstract In this paper, we define an internal binary operation between functions called fractal convolution that when applied to a pair of mappings generates a fractal function. This is done by means of a suitably defined iterated function system. We study in detail this operation in 𝓛p spaces and in sets of continuous functions in a way that is different from the previous work of the authors. We develop some properties of the operation and its associated sets. The lateral convolutions with… 
2 Citations

Figures from this paper

Convolved fractal bases and frames
Based on the theory of fractal functions, in previous papers, the first author introduced fractal versions of functions in $${\mathcal {L}}^p$$ -spaces, associated fractal operator and some related
On Some Generalizations of B-Splines
In this article, we consider some generalizations of polynomial and exponential B-splines. Firstly, the extension from integral to complex orders is reviewed and presented. The second generalization


Fractal Functions of Discontinuous Approximation
A procedure for the definition of discontinuous real functions is developed, based on a fractal methodology. For this purpose, a binary operation in the space of bounded functions on an interval is
The methodology of fractal sets generates new procedures for the analysis of functions whose graphs have a complex geometric structure. In the present paper, a method for the definition of fractal
Fundamental Sets of Fractal Functions
Abstract Fractal interpolants constructed through iterated function systems prove more general than classical interpolants. In this paper, we assign a family of fractal functions to several classes
Local Fractal Functions and Function Spaces
We introduce local iterated function systems (IFSs) and present some of their basic properties. A new class of local attractors of local IFSs, namely local fractal functions, is constructed. We
Accurate relationships between fractals and fractional integrals: New approaches and evaluations
Abstract In this paper the accurate relationships between the averaging procedure of a smooth function over 1D- fractal sets and the fractional integral of the RL-type are established. The numerical
Abstract Hardin and Massopust[1] introduced a class of fractal interpolation functions and calculated their Bouligand dimensions. This paper deals with the non-differentiability of these functions
Fractal dimension of Riemann-Liouville fractional integral of 1-dimensional continuous functions
  • Yongshun Liang
  • Mathematics
    Fractional Calculus and Applied Analysis
  • 2018
Abstract The present paper investigates fractal dimension of fractional integral of continuous functions whose fractal dimension is 1 on [0, 1]. For any continuous functions whose Box dimension is 1
Holder property of fractal interpolation function
AbstractThe purpose of this paper is to prove a Hölder property about the fractal interpolation function L(x), ω(L,δ)=O(δq, and an approximate estimate $$|f - L \leqslant 2\{ w(h) + \frac{{||f||}}{{1
Fractal functions and interpolation
Let a data set {(xi,yi) ∈I×R;i=0,1,⋯,N} be given, whereI=[x0,xN]⊂R. We introduce iterated function systems whose attractorsG are graphs of continuous functionsf∶I→R, which interpolate the data
Fractal Functions, Fractal Surfaces, and Wavelets
(Subchapter Titles): I. Foundations. Mathematical Preliminaries: Analysis and Topology. Probability Theory. Algebra. Construction of Fractal Sets: Classical Fractal Sets. Iterated Function Systems.